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Chapter 3: Problem 42

Solve the exponential equation algebraically. Approximate the result to three decimal places. $$4^{-3 t}=0.10$$

Short Answer

Expert verified
The solution to the equation \(4^{-3 t}=0.10\) is \(t \approx 0.722\).

Step by step solution

01

Express the equation in logarithmic form

The given equation is \(4^{-3 t}=0.10\). This can be expressed in logarithmic form as \(-3t = \log_{4}(0.10)\) such that \(-3t = \frac{\ln(0.10)}{\ln(4)}\), where both the numerator and denominator are computed using the natural logarithm (ln).
02

Isolate the variable

We can then proceed to isolate \(t\) by dividing both sides of the equation by \(-3\) as follows: \(t = -\frac{\ln(0.10)}{3 \ln(4)}\).
03

Compute the result

Substitute the natural logarithm values into the equation to compute the value of \(t\) using a calculator. Round off the value to three decimal places.

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